System Model

1.8 Thesis Organization

2.1.1 System Model

The system considered here consists of a multi-antenna relay (r), placed between source (s) and destination (d). Relay node is assumed to be equipped with one transmitting antenna andn receiving antennas¹. Due to half-duplex nature of relay, the source transmits to the relay and destination in first hop (i.e. s →r ands →d). A multi-antenna relay receives signal from the source through multiple links and either coherently combines them or selects the best link. If received signal is above the threshold of the decoder (active mode), the message is assumed to be decoded and re-transmitted to the destination (i.e. r →d). If this condition is not fulfilled (inactive mode), only direct path is used.

Signals received via direct path and via relay path are then either coherently combined or best one of them is selected by destination. In our system model it is assumed that equal power is allocated to s, r, identical noise power at randd as well as channel parameters are available at the receiver. Fading envelop is Rayleigh distributed so received SNR is exponentially distributed, hence PDF of received SNR betweens→dorr→d, can be written as [YA05]

f_γij(γ) =λ_ijexp (−λ_ijγ), (2.1.1) where, i∈ {s, r_o}, j ∈ {d}, λ_ij = (2`<sub>ij</sub>/`_sd)^ψ

.

ω, ψ is path loss exponent, `<sub>ij</sub> represents distance between nodeiandj, which is normalized by reference distance`sd/2,γis running variable of received SNR, ω is the SNR at reference point. In this system model independent and identically distributed (i.i.d.) fading betweens→r (i.e. ∀λ_srk = ¯λ_sr) has been assumed. So PDF of received SNR atr can

1In this chapter, symbolrorepresents transmitting antenna ofrandrkrepresents thek^threceiving antenna ofr.

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2.1. MULTI-ANTENNA RELAY SYSTEM WITH TYPE-1 OPERATION 18 be written as [YA05] and [KK08b, Eq.(6)]

f_γsr(γ) =





λ¯ⁿsrγⁿ⁻¹ (n−1)! exp¡

−λ¯_srγ¢

f or MRC

n¯λsrexp(^−¯λsrγ)

[^1−exp(^−¯λsrγ)]¹⁻ⁿ f or SC. (2.1.2) Average received SNR atddue tos→d,r→dtransmissions will be$_sd (=1/λ_sd)&$_rd(=1/λ_rd), re- spectively. Due tos→rtransmission, average received SNR atrafter SC will be$^SC_sr

µ

= 1/λ¯_srPⁿ

k=1

1/k

¶

[KM99, Eq.(8)] and average received SNR atrafter MRC will be$_sr^{M RC} ¡

=n±λ¯_sr¢

[KM99, Eq.(9)].

In first hop, if the channel condition between s →r link (after coherent addition in case of MRC or selection of best link for SC) is above a given threshold, then relay decodes the message received from the source. In this condition, mutual information (I), transmitted by the source is greater than target data rateR(spectral efficiency) [LW03]:

I =







1 2log₂

·

1 + Pⁿ

k=1

γ_srk

¸

> R f or MRC

1

2log₂[1 + max (γsrk)]> R f or SC

(2.1.3)

In equation (2.1.3), logarithm has been multiplied with1/2because such system operates in two time- slots and utilize only1/2part of channel.

Probability of relay in inactive mode

Probability that the relay is in inactive mode is given by²

P [I ≤R] =





 P

·Pn

k=1

γsrk ≤χ

¸

f or M RC P [max (γ_srk)≤χ] f or SC,

(2.1.4)

here χ = 2^2R−1 is the threshold. In our system model we assume a scenario of independent and identically distributed (i.i.d.) fading between s → r (i.e. ∀λsrk = ¯λsr). So, (2.1.4) can be evaluated with the help of (2.1.2) and [GR07, Eq.(3.381.1)]

P [I ≤R] =





Γ(^n,¯λsrχ)

(n−1)! f or MRC

£1−exp¡

−¯λ_srχ¢¤_n

f or SC, (2.1.5)

hereΓ (·,·)is upper incomplete gamma function [GR07, Eq.(8.350.1)]. Probability that relay in active mode

P [I > R] = 1−P [I ≤R]. (2.1.6)

2For avoiding the error propagation from the relays, relay decode the message if received instantaneous SNR at relay is above the threshold or the decoded data block always passes the CRC test [AY06].

2.1. MULTI-ANTENNA RELAY SYSTEM WITH TYPE-1 OPERATION 19

10 20 30 40 50 60 70 80 90

−10

−5 0 5 10 15 20 25 30

Relay positions with respect to source

Average Received SNR (dB)

Analytical Simulated

ϖ _sr^SC ϖ _rd,n=2,4 n=4

n=2 ϖ _sr^MRC

ϖ _sd,n=2,4

(a)

10 20 30 40 50 60 70 80 90

−15

−10

−5 0 5 10 15 20 25 30 35

Relay positions with respect to source

Average Received SNR (dB)

Analytical Simulated

ϖ _sr^SC ϖ _rd,n=2,4 n=4

n=2 ϖ _sr^MRC

ϖ _sd,n=2,4

(b)

10 20 30 40 50 60 70 80 90

−10

−5 0 5 10 15 20 25 30

Relay positions with respect to source

Average received SNR (dB)

Analytical Simulated

n=4

n=2 ϖ_sr^MRC

ϖ _sr^SC

ϖ _rd,n=2,4

ϖ _sd,n=2,4

(c)

10 20 30 40 50 60 70 80 90

−10

−5 0 5 10 15 20 25 30 35 40

Relay positions with respect to source

Average received SNR (dB)

Analytical Simulated

n=4

ϖ _sd,n=2,4 ϖ _rd,n=2,4 ϖ _sr^MRC

ϖ_sr^SC

n=2

(d)

10 20 30 40 50 60 70 80 90

−5 0 5 10 15 20 25 30 35

Relay positions with respect to source

Average Received SNR (dB)

Analytical Simulated

ϖ _sr^SC ϖ _rd,n=2,4

n=4 n=2

ϖ _sr^MRC

ϖ _sd,n=2,4

(e)

10 20 30 40 50 60 70 80 90

−10 0 10 20 30 40 50

Relay positions with respect to source

Average Received SNR (dB)

Analytical Simulated

ϖ _sr^SC ϖ _rd,n=2,4 n=4

n=2 ϖ _sr^MRC

ϖ _sd,n=2,4

(f)

Figure 2.2: Average received SNR of $_sr^{M RC}, $_sr^SC,$_sd and$_rd for (a)ψ = 3,ω = 0dB, (b) ψ = 4, ω = 0dB, (c)ψ = 3,ω = 3dB, (d)ψ = 4,ω = 3dB, (e)ψ = 3,ω = 7dB, (f)ψ = 4,ω = 7dB, when rplaced at various locations betweensandd.

PDF of received SNR based on link condition

Let, random variable Θmodels the received SNR atd via relay link. So, the conditional PDF of received SNR, whenris not active

f_Θ|I≤R(θ) =δ(θ), (2.1.7)

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2.1. MULTI-ANTENNA RELAY SYSTEM WITH TYPE-1 OPERATION 20 whereδ(·)is the dirac delta function. For the case when the relay is active, conditional PDF of received SNR is given as

f_Θ|I>R(θ) =λ_rodexp (−λ_rodθ). (2.1.8) PDF of received SNR, whendperforms MRC

From the theorem on total probability, PDF of received SNR can be written as

f_Θ^srd(θ) =f_Θ|I≤R(θ)P [I ≤R] +f_Θ|I>R(θ)P[I > R]. (2.1.9) The MGF³off_Θ^srd(θ), through relay link is given by

M_srd(s) =P [I ≤R] +P[I > R] λ_rod s+λrod

. (2.1.10)

When channels betweens→r→dands→dare independent, MGF of equivalent link is given by

M_T(s) = M_srd(s)M_sd(s), (2.1.11)

where,M_sd(s) = λ_sd/(s+λ_sd)is MGF of direct link. From (2.1.11), PDF of total received SNR atd can be calculated, by taking inverse Laplace transform ofMT(s)

f_Θ⁽¹⁾(θ) =P [I ≤R]λ_sdexp (−λ_sdθ) +P [I > R]

µ λ_rodλ_sd λrod−λsd

¶

{exp (−λ_sdθ)−exp (−λ_rodθ)}

(2.1.12) PDF of received SNR, whendperforms SC

The MGF offsd(γ)(direct link) can be written as M_sd(s) = λ_sd

s+λ_sd. (2.1.13)

From (2.1.13), CDF of received SNR through direct link can be calculated by taking inverse Laplace transform ofM_sd(s)/s

F_γsd(γ) = 1−exp (−λ_sdγ), (2.1.14) From (2.1.10), CDF of received SNR through relay link can be calculated by taking inverse Laplace transform ofM_srd(s)/si.e.

Fγsrd(γ) =P [I ≤R] +P [I > R] [1−exp (−λrodγ)]. (2.1.15)

3M GF =R_∞

0 f_Θ^srd(θ) exp (−sθ)dθ

2.1. MULTI-ANTENNA RELAY SYSTEM WITH TYPE-1 OPERATION 21 For the case, when channels between s → r → d and s → d are assumed to be independent and d selects the best path, then PDF of total received SNR atdcan be written as

f_Θ⁽²⁾(θ) = d

dθ {F_γsd(θ)×F_γsrd(θ)}

= P [I ≤R]λ_sdexp (−λ_sdθ) +P [I > R]

×[λsdexp (−λsdθ) +λrodexp (−λrodθ) (2.1.16)

−(λsd+λrod) exp{−(λsd+λrod)θ}]